If the vectors $\widehat {\bf{i}} - 2\widehat {\bf{j}} + 3{\bf{\hat k}},\;\; - 2\widehat {\bf{i}} + 3\widehat {\bf{j}} - 4{\bf{\hat k}},\;{\rm{\lambda \;}}\widehat {\bf{i}} - \widehat {\bf{j}} + 2{\bf{\hat k}}\;$ are linearly dependent, then the value of ${\rm{\lambda }}$ is equal to

JEE MATHEMATICS MCQ: SCALAR TRIPLE PRODUCT

1 Easy

1 of 1

Application +4 | -1

If the vectors

\widehat \mathbf{i} - 2\widehat \mathbf{j} + 3\mathbf{\hat k},\;\; - 2\widehat \mathbf{i} + 3\widehat \mathbf{j} - 4\mathbf{\hat k},\;\mathrm{\lambda \;}\widehat \mathbf{i} - \widehat \mathbf{j} + 2\mathbf{\hat k}\;

are linearly dependent, then the value of

\mathrm{\lambda }

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because linearly dependent vectors have a scalar triple product of zero, leading to lambda = 0.

Practice All Chapter MCQs →

\vec r\cdot\vec a = \vec r\cdot\vec b = \vec r\cdot\vec c = 0

\vec r,

\left[ {\vec a\vec b\vec c} \right],

\frac{{x - 3}}{2} = \frac{{y - 2}}{3} = \frac{{z - 1}}{{ - 1}}

\frac{{x + 3}}{2} = \frac{{y - 6}}{1} = \frac{{z - 5}}{3},

\mathbf{\vec a} = \widehat \mathbf{i} - \mathbf{\hat k},\;\mathbf{\vec b} = x\widehat \mathbf{i} + \widehat \mathbf{j} + \left( {1 - x} \right)\mathbf{\hat k}

\mathbf{\vec c} = y\widehat \mathbf{i} + x\widehat \mathbf{j} + \left( {1 + x - y} \right)\mathbf{\hat k}.

\left[ {\mathbf{\vec a},\;\mathbf{\vec b},\;\mathbf{\vec c}} \right]

\vec{a}, \vec{b}

((\vec{a} \times \vec{b}) \times \vec{a}) \cdot((\vec{b} \times \vec{a}) \times \vec{b})

a \cdot(a \times b)=

\mathbf{r}=\mathbf{b}+\lambda \mathbf{c}